Reasoning with Deterministic and Probabilistic graphical models Class 2: Inference in Constraint Networks Rina Dechter

Transcription

1 Reasoning with Deterministic and Probabilistic graphical models Class 2: Inference in Constraint Networks Rina Dechter

2 Road Map n Graphical models n Constraint networks Model n Inference n Search n Probabilistic Networks

3 Constraint Networks Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: A B, A D, D E, etc. A Constraint graph A B red green red yellow green red green yellow yellow green yellow red A B C D G E F B A D E G F C

4 Constraint Satisfaction Tasks Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: A B, A D, D E, etc. Are the constraints consistent? Find a solution, find all solutions Count all solutions Find a good (optimal) solution

5 Constraint Network n A constraint network is: R=(X,D,C) n X variables n D domain n C constraints X = { X 1,..., X n } D = { D,..., Dn}, Di = { v1,... v C = { C 1,... C } n R expresses allowed tuples C = over ( S, Rscopes ) 1 k n A solution is an assignment to all variables that satisfies all constraints (join of all relations). n Tasks: consistency?, one or all solutions, counting, optimization i i i t }

6 Crossword puzzle n Variables: x1,, x13 n Domains: letters n Constraints: words from {HOSES, LASER, SHEET, SNAIL, STEER, ALSO, EARN, HIKE, IRON, SAME, EAT, LET, RUN, SUN, TEN, YES, BE, IT, NO, US}

7 The Queen problem The network has four variables, all with domains Di = {1, 2, 3, 4}. (a) The labeled chess board. (b) The constraints between variables.

8 Constraint s representations n Relation: allowed tuples n Algebraic expression: n Propositional formula: X 1 2 Y 3 1 Z 2 3 n Semantics: by a relation X 2 + Y 10, X Y ( a b) c

9 Partial solutions Not all consistent instantiations are part of a solution: (a) A consistent instantiation that is not part of a solution. (b) The placement of the queens corresponding to the solution (2, 4, 1, 3). (c) The placement of the queens corresponding to the solution (3, 1, 4, 2).

10 Constraint Graphs (primal) Queen problem

11 Constraint Graphs: Primal, Dual and Hypergraphs A (primal) constraint graph: a node per variable arcs connect constrained variables. A dual constraint graph: a node per constraint s scope, an arc connect nodes sharing variables =hypergraph

12 Graph Concepts Reviews: Primal, Hyper and Dual Graphs n A hypergraph n Dual graphs n A primal graph

13 Propositional Satisfiability ϕ = {( C), (A v B v C), ( A v B v E), ( B v C v D)}.

14 Constraint graphs of 3 instances of the Radio frequency assignment problem in CELAR s benchmark

15 Operations With Relations n Intersection n Union n Difference n Selection n Projection n Join n Composition

16 Combination Local Functions f g n Join : = n Logical AND: f g =

17 Global View of the Problem C1 C2 Global View = What about counting? TASK true is 1 false is 0 logical AND? Number of true tuples Sum over all the tuples

18 Road Map n Graphical models n Constraint networks Model n Inference n Variable elimination: n Tree-clustering n Constraint propagation n Search n Probabilistic Networks

19 Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) = = Bucket E: E D, E C Bucket D: D A Bucket C: C B Bucket B: B A Bucket A: D = C A C B = A contradiction Complexity : O(n exp(w * w - induced width * ))

20 The Idea of Elimination eliminating E C R DBC = RED REB REC DBC D RDBC B Eliminate variable E joinand project value assignment 3

21 Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) {1,2} D A E {1,2} { 1,2} {1,2,3 } C B {1,2} Bucket( E) : Bucket( D) : Bucket( C) : Bucket( B) : Bucket( A) : Bucket( A) : Bucket( D) : Bucket( C) : Bucket( B) : Bucket( E) : E D, D A C B B A RA RDCB RACB RAB A D, A B D E RDB C B, C E B E RDBE RCBE, RE * Complexity : O(n exp(w (d))), * w (d) - induced widthalong ordering d E C, E B E D C B A A D C B E

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23 Adaptive Consistency, Bucket-elimination Initialize: partition constraints into bucket 1,...,bucket n For i=n down to 1 along d // process in reverse order for all relations R do // join all relations,..., 1 R m bucket and project-out i R new ( R R new ( X ) j j If is not empty, add it to where k is the largest variable index in Else problem is unsatisfiable Return the set of all relations (old and new) in the buckets i ) X i bucket k, k < i, R new

24 Properties of bucket-elimination (adaptive consistency) n Adaptive consistency generates a constraint network that is backtrack-free (can be solved without deadends). n The time and space complexity of adaptive consistency along ordering d is exponential in w*. n Therefore, problems having bounded induced width are tractable (solved in polynomial time). n trees ( w*=1), n series-parallel networks ( w*=2 ), n and in general k-trees ( w*=k).

25 Solving Trees (Mackworth and Freuder, 1985) Adaptive consistency is linear for trees and equivalent to enforcing directional arc-consistency (recording only unary constraints)

26 Tree Solving is Easy T Z 1,2,3 X < 1,2,3 1,2,3 R Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

27 Tree Solving is Easy T Z 1,2,3 X < 1,2,3 1,2,3 R Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

28 Tree Solving is Easy T Z R 1,2,3 X < 1,2 1,2 Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

29 Tree Solving is Easy T Z R 1,2,3 X < 1,2 1,2 Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

30 Tree Solving is Easy T Z R 1 X < 1,2 1,2 Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

31 Tree Solving is Easy T Z R 1 X < 2 2 Y < 1,2,3 1,2,3 1,2,3 1,2,3 S U

32 Tree Solving is Easy T Z R 1 X < 2 2 S Y < U

33 Tree Solving is Easy T Z R 1 X < 2 2 S Y < U Constraint propagation Solves trees in linear time

34 Road Map n Graphical models n Constraint networks Model n Inference n Variable elimination: n Tree-clustering n Constraint propagation n Search n Probabilistic Networks

35 Tree Decomposition A B R(a), R(b,a), R(c,a,b) R(d,b), R(f,c,d) R(e,b,f), R(g,e,f) 1 2 A B C R(a), R(b,a), R(c,a,b) BC B C D F R(d,b), R(f,c,d) Each function in a cluster C F D E 3 BF B E F R(e,b,f) Satisfy running intersection property G 4 EF E F G R(g,e,f)

36 Cluster Tree Elimination sep(2,3)={b,f} elim(2,3)={c,d} A B C R(a), R(b,a), R(c,a,b) BC B C D F R(d,b), R(f,c,d),h(1,2)(b,c) BF B E F R(e,b,f), h(2,3)(b,f) h( 1,2) ( b, c) = a R( a) R( b, a) R( c, a, b) h( 2,3) ( b, f ) = c, d R( d, b) R( f, c, d ) h(1,2 ) ( b, c) 4 EF E F G R(g,e,f) Computes the minimal domains

37 CTE: Cluster Tree Elimination A 1 ABC h( 1,2) ( b, c) = a R( a) R( b, a) R( c, a, b) C B D E 2 BC BCDF BF h( 2,1) ( b, c) = d, f R( d, b) R( f, c, d ) h(3,2) ( b, f ) h( 2,3) ( b, f ) = c, d R( d, b) R( f, c, d ) h(1,2 ) ( b, c) F 3 BEF h( 3,2) ( b, f ) = e R( e, b, f ) h(4,3) ( e, f ) Time: O ( exp(w*+1 )) Space: O ( exp(sep)) G 4 EF EFG h( 3,4) ( e, f ) = b R( e, b, f ) h(2,3) ( b, f ) h( 4,3) ( e, f ) = g R( g, e, f ) Computes the minimal domains

38 Tree decompositions A B C R(a), R(b,a), R(c,a,b) A tree decomposition for R =< X,D,C > is a triple < T, χ, ψ >, where T = (V,E) is a tree and χ and ψ are labelings over vertex v V χ(v) X and ψ(v) C satisfying : 1. For each function C C ψ(v) and scope(c i 2. For each variable X i i C there is exactly one vertex such that i ) χ(v) X the set {v V X connected subtree (running intersection property) A i χ(v)} forms a BC B C D F R(d,b), R(f,c,d) BF B E F R(e,b,f) EF Belief network C B F D E E F G R(g,e,f) Tree decomposition G

39 Induced-width and Tree-width Induced-width Of ordering D E C A B DCBA EDCB Tree-width =3 ADB DBE Tree-width =2 CBE Tree-width of a graph = smallest cluster in a cluster-tree Path-width of a graph = smallest cluster in a cluster-path

40 Road Map n Graphical models n Constraint networks Model n Inference n Variable elimination: n Tree-clustering n Constraint propagation n Search n Probabilistic Networks

41 Sudoku Approximation: Constraint Propagation Constraint Propagation Inference Variables: empty slots Domains = {1,2,3,4,5,6,7,8,9} Constraints: 27 all-different Each row, column and major block must be alldifferent Well posed if it has unique solution: 27 constraints

42 Approximating Inference: Local Constraint Propagation n Problem: bucket-elimination/tree-clustering algorithms are intractable when induced width is large n Approximation: bound the size of recorded dependencies, i.e. perform local constraint propagation (local inference)

43 From Global to Local Consistency

44 Arc-consistency A binary constraint R(X,Y) is arc-consistent w.r.t. X is every value In x s domain has a match in y s domain. RX = {1,2,3}, RY = {1,2,3}, constraint X < Y Only domains are reduced: R X X RXY DY

45 Arc-consistency 1 X, Y, Z, T 3 X < Y Y = Z T < Z X T X 1, 2, 3 Y 1, 2, 3 1, 2, 3 1, 2, 3 T < < Z =

46 Arc-consistency 1 X, Y, Z, T 3 X < Y Y = Z T < Z X T X T < Y 1 3 = 2 3 < R X Z X RXY DY

47 From Arc-consistency to relational arc-consistency A B 1 1 n Sound 2 3 A < B 2 3 n Incomplete n Always converges (polynomial) < = D 1 2 D < C C

48 Relational Distributed Arc- Consistency Primal Dual A B A B 2 3 AB B BC A C D D C C AD D DC

49 AC-3 O( ek 3 )

50 Arc-consistency Algorithms O( nek n AC-1: brute-force, distributed n AC-3, queue-based O( ek 3 ) n AC-4, context-based, optimal n AC-5,6,7,. O( ek 2 Good in special cases n Important: applied at every node of search n (n number of variables, e=#constraints, k=domain size) n Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985) 3 ) )

51 Path-consistency n A pair (x, y) is path-consistent relative to Z, if every consistent assignment (x, y) has a consistent extension to z.

52 Example: path-consistency

53 Path-consistency Algorithms n Apply Revise-3 (O(k^3)) until no change R R π ( R D Rkj n Path-consistency ij ij (3-consistency) ij ik k adds binary constraints. n PC-1: n PC-2: O( n 5 k 5 ) n PC-4 optimal: O( n 3 k 5 ) O( n 3 k 3 ) )

54 Local i-consistency i-consistency: Any consistent assignment to any i-1 variables is consistent with at least one value of any i-th variable

55 Gausian and Boolean Propagation, Resolution n Linear inequalities x + y + z 15, z n Boolean constraint x, y propagation, unit resolution ( A B C),( B) ( A C)

56 Directional Resolution ó Adaptive Consistency * bucket i = O(exp( w )) DR time and space : O( n exp( w * ))

57 Directional i-consistency D A E C B D C B E D C E B D C B E E : D : C : B : E D, E C, E D C, D A C B A B B Adaptive R DCB R d-path, DC R DB R CB d-arc R D R C R D A :

58 Greedy Algorithms for Induced-Width n Min-width ordering n Min-induced-width ordering n Max-cardinality ordering n Min-fill ordering n Chordal graphs n Hypergraph partitionings (Project: present papers on induced-width, run algorithms for inducedwidth on new benchmarks )

59 Min-width ordering Proposition: algorithm min-width finds a min-width ordering of a graph Complexity:? O(e)

60 Greedy orderings heuristics min-induced-width (miw) input: a graph G = (V;E), V = {v1; :::; vn} output: A miw ordering of the nodes d = (v1; :::; vn). 1. for j = n to 1 by -1 do 2. r ß a node in V with smallest degree. 3. put r in position j. 4. connect r's neighbors: E ß E union {(vi; vj) (vi; r) in E; (vj ; r) 2 in E}, 5. remove r from the resulting graph: V ß V - {r}. Theorem: A graph is a tree iff it has both width and induced-width of 1. min-fill (min-fill) input: a graph G = (V;E), V = {v1; :::; vn} output: An ordering of the nodes d = (v1; :::; vn). 1. for j = n to 1 by -1 do 2. r ß a node in V with smallest fill edges for his parents. 3. put r in position j. 4. connect r's neighbors: E ß E union {(vi; vj) (vi; r) 2 E; (vj ; r) in E}, 5. remove r from the resulting graph: V ß V {r}.

61 Example We see again that G in the Figure (a) is not chordal since the parents of A are not connected in the maxcardinality ordering in Figure (d). If we connect B and C, the resulting induced graph is chordal.

62 Cordal Graphs; Max-Cardinality Ordering n A graph is cordal if every cycle of length at least 4 has a chord n Finding w* over chordal graph is easy using the max-cardinality ordering n If G* is an induced graph it is chordal n K-trees are special chordal graphs. n Finding the max-clique in chordal graphs is easy (just enumerate all cliques in a max-cardinality ordering

63 Max-cardinality ordering

64 Graph Concepts: Hypergraphs and Dual Graphs n A hypergraph is H = (V,S), V= {v_1,..,v_n} and a set of subsets Hyperegdes: S={S_1,..., S_l }. n Dual graphs of a hypergaph: The nodes are the hyperedges and a pair of nodes is connected if they share vertices in V. The arc is labeled by the shared vertices. n A primal (Markov, moral) graph of a hypergraph H = (V,S) has V as its nodes, and any two nodes are connected by an arc if they appear in the same hyperedge. n Factor Graphs..

65 Which greedy algorithm is best? n MinFill, prefers a node who add the least number of fill-in arcs. n Empirically, fill-in is the best among the greedy algorithms (MW,MIW,MF,MC) n Complexity of greedy orderings? n MW is O(?), MIW: O(?) MF (?) MC is O(mn)

66 Road Map n Graphical models n Constraint networks Model n Inference n Variable elimination: n Tree-clustering n Constraint propagation n Search n Probabilistic Networks